Combinations: Selections Without Order

A combination is a selection of objects where order does not matter — only which objects are chosen, not the order in which they are picked. Choosing $\{A, B\}$ is the same combination as $\{B, A\}$.

This is the crucial contrast with a permutation. Each group of $r$ selected objects can itself be arranged in $r!$ orders, so the number of permutations is $r!$ times the number of combinations. Dividing gives:

Example: the number of ways to choose a committee of $3$ people from $5$ is $^{5}C_{3} = \dfrac{5!}{3!\,2!} = \dfrac{120}{6 \times 2} = 10$. (By contrast, arranging $3$ of the $5$ in definite posts would be $^{5}P_{3} = 60 = 10 \times 3!$ — six times as many, one for each ordering of a chosen trio.)